Mathematics Preparation for UW-Madison

Studying Mathematics

Mathematics at the college level is a rich subject, full of interesting ideas and surprising applications. Students with real interests and abilities in mathematics or kindred subjects, such as physics or theoretical computing, should be thinking about eventually studying mathematics at this level. Their high school preparation should entail taking as much mathematics as they can and also reading books that will suggest some of the power and interest of mathematics, such as Mathematics Today by Lynn Steen or Wheels, Life and Other Mathematical Amusements by Martin Gardner. A fascinating reading list is given at the end of Metamagical Themas by Douglas Hofstadter.

Most students enroll in mathematics classes for a utilitarian purpose-they wish to prepare themselves for quantitative work in programs other than mathematics. For students who choose to study mathematics at the university, four years of high school math is immensely beneficial. It is essential to alert students to the pervasive quantitative nature of virtually all modern learning and analysis. Thus, the level of mathematical skill necessary for most fields of study increases daily.

The minimal entrance requirement in mathematics for UW-Madison is three years of high school mathematics. Further, this is expected to cover certain specific topics, centered on algebra, geometry, and trigonometry. Some courses which do carry high school credit may not count toward this entrance requirement.

Students who wish to get off to a good start in college should take more than that minimal requirement. Certain courses and majors require mathematics prerequisites. For example, students who expect to complete certain scientific or technological majors should view calculus as basic preparation. The time needed to complete precalculus courses may add an extra semester or an extra year to complete those programs. In addition, even if not majoring in a technical area, a student will have to meet a Quantitative Reasoning requirement before being allowed to graduate. There are various courses which may be taken to meet this requirement, and they in turn expect preparation in mathematics. A student who comes in prepared to start with calculus will be exempt from one and possibly all parts of the QR course requirements.

Both colleges and specific majors within colleges may have math requirements beyond admissions levels. Students are responsible for learning of such requirements as soon as they begin the advising process. It is important to plan ahead to take prerequisites as necessary.

Competency Levels

All entering freshmen at UW-Madison are placed in mathematics courses according to their demonstrated "competency levels"- a determination made on the basis of high school work and performance on the University of Wisconsin System Mathematics Placement Test. The competency level description appears at the end of this section (p. 9), showing various areas of college study, with the appropriate competency level for each. The competency levels are minimum, intermediate, and advanced. It is expected that a student working conscientiously in a mathematics program should reach minimum competence after two years of high school study, intermediate after three, and advanced after four.

For students entering UW-Madison in the fall of 1998, the last year for which comprehensive data are available, most had three or four or even five or more years of high-school mathematics. The few who entered with only two units all placed into remedial courses and would have to take extra mathematics courses in order to meet graduation requirements. Only 10% had as little as three years: Four or more has become the norm for entering students.

Students entering with only three years did vary in the placement level they achieved: Approximately 29% tested as minimally competent or below, resulting in placement into courses which would require additional math to be taken no matter what major is selected. The largest group of these students, 64%, ranked at intermediate competence and so placed into College Algebra or equivalent courses: Success in that course would meet the minimal mathematics requirements for graduation from the University, but not be sufficient for most mathematics-using majors.

Students entering with four years of high-school math were about 46% of the entering class. But many did not bring with them from high-school all the material they should have learned in those years. The largest fraction of these students, about 60%, still rated only intermediate competence and placed into College Algebra, despite having supposedly covered much of that material in their high-school courses. Roughly 7% of these students did not even do that well, achieving only intermediate competence or not even that, and so going into courses needed to prepare them for College Algebra. About 25% of the students with four years of high school math were prepared to start with calculus, which most UW-Madison faculty view as the place students should expect to begin their college mathematics.

More than 43% of the entering students in 1998 had five or more years of high-school mathematics. About 58% of those students were prepared to begin their college mathematics with calculus and so were not set back in their college careers. But approximately 28% of these students with extensive high-school preparation still placed at the intermediate competence level or below, i.e. into College Algebra or lower courses. About 13% of these students did not have to take College Algebra or anything lower, but did need additional trigonometry before being ready for calculus.

Part of this gap between the ideal and the actual may result from high school courses that do not cover the mathematical material central to college preparation. Another part of this gap is unquestionably the result of students who do not learn the central material at a level high enough for college preparation; more on this topic appears in the next section, "Learning Mathematics." Finally there is the problem of lack of retention. Students who do not take mathematics during their senior year forget significant parts of the mathematics they once knew, and they enter college at a level below the best they have attained.

Topics to Study

Geometry, algebra/pre-calculus, and trigonometry are the core of the college preparatory program. Geometry must be studied at the level of learning associated with a college-track course; basic geometry courses are not sufficient. Algebra needs to be mastered thoroughly-students wishing to be well prepared will take two algebra courses. Precalculus refers to material such as function notation, logarithms and exponentials, and analytic geometry. Calculus itself fits into the core if taught in a version generating college credit (e.g., a course affiliated with the Advanced Placement Program of The College Board); other versions do not. A course in finite or discrete mathematics contributes to college preparation, as does the ability to use a scientific calculator. Computer literacy is helpful in a general way but does not contribute directly to work in entry-level mathematics courses. None of the mathematics courses an entering student at UW-Madison might take will expect familiarity with any particular calculator or computer technology. Some courses may make use of such technology, in which case they will be self-contained and let you know how to use it. Very few mathematics courses require a student to use such technology.

Learning Mathematics

Just as important as the lists of topics covered in college preparatory mathematics is the level of understanding attained. Students who see mathematics only as a collection of rules to be mechanically applied in stereotyped situations are not mathematically prepared. From the beginning, students should learn to apply mathematical skills in a variety of contexts. For example, after studying quadratics they should solve 4-x²=16x as readily as $x2+16x-4=0$. At a somewhat later stage, they should see that solving $4-y4=16y2$ is done by solving quadratics. From penetrating these thin disguises, they should progress by separating a problem into simpler pieces, finding a way to deal with the pieces, and putting the results together to solve the original problem. For example, given three points P, Q, and R in the plane, the student must determine whether there is a point equidistant from all three. Looking just at the points equidistant from P and Q, we note that they must lie on the perpendicular bisector of segment PQ. Now looking at the points equidistant from Q and R, we find they lie on the perpendicular bisector of QR. The two bisectors either intersect at a single point or are parallel. The student finds the solution to be that there is a unique point equidistant from P, Q, and R unless these three points lie on a line. Being prepared in geometry means being able to do this kind of analysis.

The problems in the paragraph above are not ones for which students should prepare specifically. Rather, they are examples of a very large variety of problems accessible without specific preparation to students whose learning has been at the desired level. Of course, the best way for students to reach this level is for them to do a large variety of problems consistently throughout their high school mathematics work. Being prepared in mathematics also means knowing what kinds of problems you can solve and when you have sufficient data to determine an answer. This kind of general understanding is essential for "problem solving" and "word problems." Finally, students maintain their mathematical learning best if they have acquired it in a logical structure; such a logical structure permits them to retain the many details necessary to use mathematics successfully.

Mathematics and Choice of College Discipline

We return to an earlier point: students who enter the university with minimal mathematical preparation are at a serious disadvantage in choosing and completing some majors. This disadvantage, unfortunately, goes unrecognized at first because math skills are often a "hidden prerequisite." Many programs that have no stated mathematics requirements do require courses that make crucial use of math skills. In the social sciences, such requirements do require courses that make crucial use of math skills. In the social sciences, such requirements will be in the area of statistics (often under a guise of "Quantitative Methods"). The chemistry courses required for biological and health science majors have "hidden" math prerequisites, with intermediate and advanced courses requiring two semesters of calculus. Alert and well-advised students will take such needs into account in planning their freshman year. They will recognize that the alternatives are to limit the field of possible college majors to those for which they are already prepared, or to begin at once to get the necessary preparation, even if this adds a semester or two to their college careers. Good mathematical preparation is like an automobile; it is not necessary for survival in college, but it makes students much more mobile in their choice of college majors. The list below illustrates the math competencies students will need for completing programs in various areas at UW-Madison. Definitions of competency follow the table.

Note: Students may meet these levels during the course of other undergraduate work, but doing so detracts from work in the program itself.

Definition of Mathematical Competencies at UW-Madison

1. Minimum mathematical competence

From algebra and arithmetic:

From geometry:

2. Intermediate mathematical competence

The topics of level 1, together with

3(a). Advanced mathematical competence, Algebra:

The topics of levels 1 and 2, together with:

3(b). Advanced mathematical competence: Trigonometry

The topics of levels 1 and 2, together with: